Affine Isoperimetric Type Inequalities for Static Convex Domains in Hyperbolic Space

Abstract

In this paper, the notion of hyperbolic ellipsoids in hyperbolic space is introduced. Using a natural orthogonal projection from hyperbolic space to Euclidean space, we establish affine isoperimetric type inequalities for static convex domains in hyperbolic space. Moreover, equality of such inequalities is characterized by these hyperbolic ellipsoids.

Keywords

Static convex / Blaschke-Santalό inequality / affine isoperimetric inequality / orthogonal projection / hyperbolic space

Acknowledgements

This work is supported by National Key Research and Development Program of China (Grant No. 2021YFA1001800), National Natural Science Foundation of China (Grant Nos. 12101027 and 12471047) and the Fundamental Research Funds for the Central Universi-ties. The research leading to these results is part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 101001677).

References

[1] Agostiniani V, Fogagnolo M, Mazzieri L. Minkowski inequalities via nonlinear potential theory. Arch. Rational Mech. Anal., 2022, 244: 51-85.

[2] Andrews B, Chen X, Wei Y. Volume preserving flow and Alexandrov-Fenchel type in-equalities in hyperbolic space. J. Eur. Math. Soc., 2021, 23(7): 2467-2509.

[3] Andrews B, Hu Y, Li, H. Harmonic mean curvature flow and geometric inequalities. Adv. Math., 2020, 375: 107393.

[4] Blaschke W. Über affine geometrie I: isoperimetrische eigenschaften von ellipse and el-lipsoid. Ber. Verh. Sächs. Akad. Wiss. Leipzig. Math.-Phys. KI., 1916, 68: 217-239.

[5] Blaschke W. Vorlesungen Über Differentialgeometrie II: Affine Differentialgeometrie. Springer, Berlin, 1923.

[6] Bourgain J, Milman V D. New volume ratio properties for convex symmetric bodies in $\mathbb{R}^{b} $. Invent. Math., 1987, 88: 319-340.

[7] Brendle S. Constant mean curvature surfaces in warped product manifolds. Publ. Math. l’IHÉS, 2013, 117: 247-269.

[8] Brendle S, Hung P K, Wang M T. A Minkowski-type inequality for hypersurfaces in the Anti-deSitter-Schwarzschild manifold. Comm. Pure Appl. Math., 2016, 69(1): 124-144.

[9] Brendle S, Wang M T. A Gibbons-Penrose inequality for surfaces in Schwarzschild space-time. Commun. Math. Phys., 2014, 330: 33-43.

[10] Deicke A. Über die Finsler-Räume mit $A_{i}=0 $. Arch. Math., 1953, 4: 45-51.

[11] Lima L L, Girao F. An Alexandrov-Fenchel-type inequality in hHyperbolic space with an application to a Penrose inequality. Ann. Henri Poincare, 2016, 17: 979-1002.

[12] Gao F, Hug D, Schneider R. Intrinsic volumes and polar sets in spherical space. Math. Notae., 2001, 41: 159-176.

[13] Ge Y, Wang G, Wu J. Hyperbolic Alexandrov-Fenchel quermassintegral inequalities II. J. Diff. Geom., 2014, 98(2): 237-260.

[14] Ge Y, Wang G, Wu J. The GBC mass for asymptotically hyperbolic manifolds. Math. Z, 2015, 281: 257-297.

[15] Gibbons G W. Collapsing shells and the isoperimetric inequality for black holes. Class. Quantum Grav., 1997, 14: 2905-2915.

[16] Hu Y, Li H. Geometric inequalities for static convex domains in hyperbolic space. Trans. Amer. Math. Soc., 2022, 375(8): 5587-5615.

[17] Hu Y, Li H. Geometric inequalities for hypersurfaces with nonnegative sectional curva-ture in hyperbolic space. Calc. Var. Partial Differential Equations. 2019, 58(2): 55.

[18] Hu Y, Li H. Blaschke-Santaló type inequalities and quermassintegral inequalities in hy-perbolic space. Adv. Math., 2023, 413: 108826.

[19] Hu Y, Li H, Wei Y. Locally constrained curvature flows and geometric inequalities in hyperbolic space. Math. Ann., 2022, 382(3-4): 1425-1474.

[20] Hu Y, Wei Y, Zhou T. A Heintze-Karcher type inequality in hyperbolic space. J. Geom. Anal., 2024, 34(4): 113.

[21] Hug D. Curvature relations and affine surface area for a general convex body and its polar. Results Math., 1996, 29(3-4): 233-248.

[22] Li H, Wan Y. The Christoffel problem in the hyperbolic hyperplane. Adv. in Appl. Math., 2023, 150: 102557.

[23] Li H, Wei Y, Xiong C-W. A geometric inequality on hypersurface in hyperbolic space. Adv. Math., 2014, 253: 152-162.

[24] Li H, Xu B. A class of weighted isoperimetric inequalities in hyperbolic space. Proc. Amer. Math. Soc., 2023, 151(5): 2155-2168.

[25] Li H, Xu B. Hyperbolic p-sum and horospherical p-Brunn-Minkowski theory in hyper-bolic space. ArXiv:2211.06875.

[26] Li J, Xia C. An integral formula and its applications on sub-static manifolds. J. Diff. Geom., 2019, 113: 493-518.

[27] Lutwak E. The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas. Adv. Math., 1996, 118(2): 244-294.

[28] Mars M, Soria A. Geometry of Normal Graphs in Euclidean Space and Applications to the Penrose Inequality in Minkowski. Ann. Henri Poincaré, 2014, 15: 1903-1918.

[29] Petty C M. Affine isoperimetric problems. Ann. N.Y. Acad. Sci., 1985, 440: 113-127.

[30] Qiu G, Xia C. A generalization of Reilly’s formula and its applications to a new Heintze-Karcher type inequality. Int. Math. Res. Not., 2015, 2015(17): 7608-7619.

[31] Reilly R C. Geometric applications of the solvability of Neumann problems on a Rie-mannnian manifold. Arch. Ration. Mech. Anal., 1980, 75(1): 23-29.

[32] Santaló L A. Un invariante afin para los cuerpos convexos del espacio de n dimensiones. Portugaliae Math., 1949: 155-161.

[33] Scheuer J, Xia C. Locally constrained inverse curvature flows. Trans. Amer. Math. Soc., 2019, 372(10): 6771-6803.

[34] Schneider R. Ency-clopedia of Mathematics and its Applications. Cambridge University Press, Convex Bodies: the Brunn-Minkowski Theory, Second expanded ed.ed., Cambridge, 2014.

[35] Schütt C, Werner E. Surface bodies and p-affine surface area. Adv. Math., 2004, 187: 98-145.

[36] Wang G, Xia C. Isoperimetric type problems and Alexandrov-Fenchel type inequalities in the hyperbolic space. 2014, 259: 532-546.

[37] Wang M-T, Yau S-T. Quasilocal mass in general relativity. Phys. Rev. Lett., 2009, 102(2): 021101.

[38] Wang M-T, Yau S-T. Isometric embeddings into the Minkowski space and new quasi-local mass. Commun. Math. Phys., 2009, 288(3): 919-942.

[39] Werner E, Ye D. New L_p affine isoperimetric inequalities. Adv. Math., 2008, 218: 762-780.

[40] Xia C. A Minkowski type inequality in space forms. Calc. Var., 2016, 55(4): 1-8.